Recently in comics, the X-Men have been suffering quite a bit. Ever since Joss Whedon's run on Astonishing X-Men ended, it's been a struggle for Marvel to regain the spark that made the X-Men special. They tried decreasing the amount of mutants in the world, but that's just made the series about speciesism, and not human/mutant struggles.
Anyway, while the X-Men might now be relying on a cross-over with the Dark Avengers, I can't help but remember where the rampant X-Men popularity started. With Len Wein and Dave Cockrum - the X-Men changed forever, from a struggling 3ed string property, to a first class best seller. Giant X-Men #1 helped propel the X-Men to greater heights, and introduced some of the series most enduring and popular characters.
It all began with a mission to a monster Island.
The original X-Men had been caught by this living island, and this prompted Xavier to quickly recruit a new team of X-Men to help save the old ones. Storm, Nightcrawler, Colossus, Wolverine, Banshee, Sunfire, and Thunderbird where a breath of fresh air to the series. Not only did they all come equipped with fantastic new powers and looks, but each one of them came from foreign countries.
When all was said and done, the All New, All Different X-Men where formed.
Christ Claremont quickly took over the writing of the title, and slimmed down the large cast of mutants to Wolverine, Storm, Nightcrawler, Colossus, Banshee, and Cyclops. Teammate Thunderbird quickly became the first casualty, dying on one of their first missions.
The X-Men continued on, having many adventures and did something most other comics never did - continued the story line from issue to issue, creating a soap opera-like feel. Most comics at the time found this style of writing untenable, as issue sales many times depended on the reader knowing what to expect form a book each month. A continuing story line, like the X-Men portrayed, was something rare. It could have easily backfired. But in fact this is where the series and characters thrived. Chris Claremont found an excellent balance to the always changing story line, and the mandates of monthly comics, by always reminds readers, each issue key points about characters. Always explaining how their powers work and what the X-Men fought for - this helped give readers everything they needed to know when jumping into any particular issue.
The most popular character to come out of the X-Men was no doubt Wolverine. As Claremont continued to write the series, we found out, slowly, more and more about the mysterious and rouge character. It's hard to forget now, but at the time we didn't even know how his powers worked. "Where the claws part of his costume?", for instance. Much of Wolverine's character was forged within the pages of these early issues - and I think his character, in general, was probably definitively cast when Claremont and Frank Miller did their first solo series with him. (That's my opinion, at least)
You cannot mention the Claremont/Cockrum era without mentioning one of their most defining story lines, that of the Phoenix Saga. It seemed like, with all that the X-Men had done to that point, a giant space saga was the final culmination of all these adventures. Jean Grey became a prominent figure because of this story, as she merged with the Phoenix entity and helped save the universe from the villain D'Ken of the Shi'ar Empire.
After Cockrum left the series as artist, John Byrne took over and helped propel the X-Men to even greater popularity.
Byrne and Claremont completed the Phoenix Saga in one of the most spectacular ways possible. Jean Grey, now consumed by the power of the Phoenix force, went crazy - destroying planets and nearly her fellow X-Men. The Shi'ar stepped in, in response, and held that Jean Grey must die for what she did as the Dark Phoenix. In a tragic death on par with Gwen Stacy dying in Spider-Man, Jean Grey killed herself to stop herself and the Phoenix force once and for all.
After the Dark Phoenix Saga it seemed like Byrne's effect on the series finally took hold. Wolverine was put in a cool new brown costume, Kitty Pryde was introduced into the X-Men as a great new teen sidekick. Byrne and Claremont's best story line kicked off during this time, that helped define the X-Men for years to come, that of "Days of Future Past" that dramatically saw how the X-Men's mission of a peaceful co-existence could entirely fail.
Even after John Byrne left the series, the X-Men continued on. The continued evolution of the X-Men continued to be deftly handled by Chris Claremont. One of the bigger changes to the team, I think, was Storm - who's innocent personality changed, making her a fiercer fighter. While the cosmetic change didn't last forever, her commanding leadership has remained, forever strengthening the character.
Claremont continued to write the X-Men well, introducing new characters and themes into the series -- but it did eventually begin to wane. I particularly thought the Mutant Massacre story line fell flat, and the Inferno story line even worse. But it will never erase the great memories and great stories that helped make the X-Men what they are today.